Brazil star Raphinha fired Leeds ahead in the 56th minute only for Andrew Omobamidele to head Norwich level from a corner two minutes later.
But Rodrigo's long-range strike put United back in front another two minutes later and the record signing's strike proved the winning goal against the division's bottom side.
The victory, United's second Premier League win of the season, took Marcelo Bielsa's side fourth-bottom and on to ten points after ten games played, three points clear of third-bottom Burnley.
Graham Smyth, the YEP's chief football writer, hands out the scores from the triumph at Carrow Road.
13. Adam Forshaw (sub)
6 - Ran around, tried to disrupt and annoy Norwich. Wasn't on the ball as much as he'd like.
14. Tyler Roberts (sub)
N/A - Came on late. Nice pass put Raphinha away, conceded a silly free-kick in the last seconds that Norwich failed to do anything with.
15. Anthony Taylor (referee)
5 - Looked as if he would let lots go in the first few minutes but then became very fussy. Frustrated both teams. Got a few minor decisions wrong.